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ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 8 No. 2 (2019), pp. 1-10.
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⃝2019 University of Isfahan
www.ui.ac.ir
ON ALGEBRAIC GEOMETRY OVER COMPLETELY SIMPLE SEMIGROUPS
ARTEM N. SHEVLYAKOV
Communicated by Evgeny Vdovin
Abstract. We study equations over completely simple semigroups and describe the coordinate semi-groups of irreducible algebraic sets for such semisemi-groups.
1. Introduction
Many problems of semigroup theory are reduced to the similar problems of group theory. This
approach is often used for completely simple (c.s.) semigroups, since any c.s. semigroupGis isomorphic
to a disjoint union of isomorphic copies of a group G (G is called the structural group of S). Thus, S
actually inherits a big part its properties from G, and one can formulate the general principle:
(1.1) S has a propertyP ⇔G has a property P .
In this paper we study equations over c.s. semigroups. Remark that the equations over c.s.
semi-groups have more complicated view than group equations. It turns out that the principle (1.1) also
holds for equations over c.s. semigroups. For example, it was proved in [6] that the property P =
{to be an equationally Noetherian semigroup}satisfies (1.1).
Below we describe the coordinate semigroups of irreducible algebraic sets over a c.s. semigroup
S. We show that this problem is reduced to the same problem for the structural group G of S (see
Theorems 3.1,5.1). In Theorem3.1 we study equations with no constants, whereas in Theorem5.1we
consider equations with constants.
MSC(2010): Primary: 20M17; Secondary: 20M99.
Keywords: system of equations, coordinate semigroups, universal algebraic geometry. Received: 16 April 2016, Accepted: 01 September 2017.
http://dx.doi.org/10.22108/ijgt.2017.21975
Moreover, in the paper we deal with free semigroups of the variety of c.s. semigroups with abelian
structural groups and prove that such semigroups are geometrically equivalent to each other
(Theo-rem 4.2).
2. Basic Notions
The class of completely simple (c.s.) semigroups is one of the important semigroup classes. The c.s.
semigroups are described by the following famous theorem.
Theorem 2.1. For any c.s. semigroupS there exists a groupGand setsI,Λ such thatS is isomorphic to the set of triples (λ, g, i), g ∈G, λ∈Λ, i∈I. The multiplication over the triples (λ, g, i) is defined as follows
(λ, g, i)(µ, h, j) = (λ, gpiµh, j),
where piµ∈G is an element of a matrix Psuch that (1) Pconsists of |I| rows and|Λ| columns;
(2) the matrix P is (λ, i)-normalised for some λ ∈Λ, i ∈I, i.e. all elements of the λ-th row and
i-th column equal 1∈G.
Following Theorem 2.1, we denote any c.s. semigroup S by S = (G,P,Λ, I). The group G and the matrix P are called the structural group and the sandwich-matrix respectively. The elementsλ, i occurring in a triple (λ, g, i) ∈S are the first and the second indexes respectively. We will call the set Rλ ={(λ, g, i)|i∈I} ⊆S (Ci ={(λ, g, i)|λ∈Λ} ⊆S) theλ-th row(respectively, i-th column) of S. Formally, each row (column) of a c.s. semigroupS is a minimal right (respectively, left) ideal of S.
Remark 2.2. We implicitly use below that the following transformations of a c.s. semigroup S with a presentation (G,P,Λ, I) do not change the semigroup structure of S:
(1) any substitutionG7→G′, where a group G′ is isomorphic to G;
(2) a multiplication of any row (column) of P by an element g∈ G (i.e. one can normalizeP for any row and column).
(3) any swap of rows (columns) in the sandwich-matrix P.
Let ⟨P⟩ ⊆Gdenote the group generated by all entries of the sandwich-matrix P.
By Theorem2.1, any c.s. semigroupS is the disjoint union of copies of the structural groupG, i.e. all
maximal subgroups of S are isomorphic toG. Clearly, the identity elements of the maximal subgroups
are (λ, piλ−1, i), λ∈Λ, i∈I. The inversion−1 in a subgroup defined by the indexes λ, i is (λ, g, i)−1 = (λ, piλ−1g−1piλ−1, i).
The class of c.s. semigroups is a variety in the language{·,−1}, since it is defined by the identities: xx−1x=x, xx−1 =x−1x, (x−1)−1 =x, (xyx)−1(xyx) =x−1x.
Theorem 2.3. [2, 9] Let Xn = {x1, x2, . . . , xn}, Yn = {yiλ|2 ≤ i ≤ n,2 ≤ λ ≤ n} be finite sets
of letters, and In = Λn = {1,2, . . . , n}. Let F(Xn∪Yn) denote the free group generated by the set Xn∪Yn. Thus, the free c.s. semigroup Fn of ranknis defined by Fn= (F(Xn∪Yn),Pn, In,Λn), where
Pn= (piλ|i∈In, λ∈Λn),
piλ=
1, if i= 1 or λ= 1, yiλ, otherwise
and the free generators xi correspond to the triples (i, xi, i)∈ Fn.
Let CSab be the class of c.s. semigroups with abelian structural groups. Following [10], CSab is a variety, and Theorem 2.3gives the structure of the free semigroupFnab ∈CSab of rankn. Indeed,
(2.1) Fnab = (FN,Pn,Λn, In),
whereFN is isomorphic to the free abelian group generated byXn∪Yn(the setsXn, Yn,Λn, In and the
sandwich-matrix Pn were defined in Theorem 2.3). Thus, the structural group of Fnab is isomorphic to the direct sum
(2.2) ⊕
1<λ,i≤n
Zλi⊕Zn=⟨P⟩ ⊕Zn,
where Zλi are isomorphic copies of the group Z.
The following result describes the set of homomorphisms of c.s. semigroups.
Theorem 2.4. [3] Let S = (G,P,Λ, I), S′ = (G′,P′,Λ′, I′) be a c.s. semigroups and φ∈Hom(S, S′). Then, there exist mappings
(1) ¯h:I →G′, χ¯: Λ→G′,
(2) h:I →I′,χ: Λ→Λ′,
(3) a group homomorphism ω:G→G′
such that
(2.3) ω(piλ) = ¯χ(λ)ph(i)χ(λ)¯h(i)
and
(2.4) φ((λ, g, i)) = (χ(λ),¯h(i)ω(g) ¯χ(λ), h(i)).
Lemma 2.5. Let S = (G,P,Λ, I), S′ = (G′,P′,Λ′, I′) be c.s. semigroups with (1,1)-normalized P,P′. Suppose a homomorphism φ:S →S′ (2.4) maps the first column of S into the first column in S′, and the first row of S is mapped into the row column in S′. Then there exists an element a∈G′ such that
¯
h(i) =a,χ(λ) =¯ a−1 for all i∈I, λ∈Λ, and
Proof. By (2.3), for anyi∈I,λ∈Λ we have
1 =ω(1) =ω(p1λ) = ¯χ(λ)ph(1)χ(λ)¯h(1) = ¯χ(λ)p1χ(λ)¯h(1) = ¯χ(λ)¯h(1),
1 =ω(1) =ω(pi1) = ¯χ(1)ph(i)χ(1)¯h(i) = ¯χ(1)ph(i)1h(i) = ¯¯ χ(1)¯h(i).
In particular, ¯χ(1)¯h(1) = 1. Leta=h(1), thena−1 = ¯χ(λ), a= ¯h(i), and we immediately obtain (2.5).
□
Let us give the basic notions of universal algebraic geometry. All definitions below are derived from
the general notions of [4], where such definitions were formulated for an arbitrary algebraic structure in
the language with no predicates.
We consider c.s. semigroups (as algebraic structures) in the language L={,−1} with natural
inter-pretation of the functional symbols ,−1. Let X be a finite set of variables x1, x2, . . . , xn. A term of a
languageL (L-term) in variables X is one of the following expressions:
(1) a variable xi;
(2) a product t(X)s(X) of twoL-termst(X), s(X);
(3) (t(X))−1, wheret(X) is a L-term.
For example, the expressions ((x−1)−1)−1 (xy)−1,x(yz−1)−1 areL-terms.
An equationover L (L-equation) is an equality of two L-termst(X) =s(X). A system of equations
overL (L-systemorsystemfor shortness) is an arbitrary set of L-equations.
A point P = (p1, p2, . . . , pn) ∈ Sn is a solution of a system S in variables x1, x2, . . . , xn if the
substitution xi =pi reduces any equation of S to a true equality in a c.s. semigroup S. The set of all solutions of a system S in a c.s. semigroup S is denoted by VS(S). A set Y ⊆ Sn is called algebraic over a c.s. semigroup S if there exists a L-system in variablesx1, x2, . . . , xn with the solution set Y. A
nonempty algebraic setY isirreducible if it is not a proper finite union of other algebraic sets. Let us give examples of algebraic sets over a c.s. semigroup S.
(1) Y ={(x, y)|x, y belong to the same maximal subgroup}= VS(xx−1 =yy−1). (2) Y ={(x, y)|x, y belong to the same row}= VS((xy)(xy)−1=yy−1).
LetY be a non-empty algebraic set defined by a system ofL-equations. Let us define the equivalence
relation over the set of L-termsTS by
t(X)∼Y s(X)⇐⇒t(P) =s(P) for anyP ∈Y.
Denote the factor-semigroup TS/∼Y by ΓS(Y). One can prove that ΓS(Y) is a c.s. semigroup.
Follow-ing [4], ΓS(Y) is calledthe coordinate semigroup of an algebraic set Y, and the main aim of universal algebraic geometry is the description of the coordinate semigroupsof algebraic sets over a given semigroup S.
Theorem 2.6. [5] Let S, S′ be a completely simple semigroups and S is finitely generated. Then the following conditions are equivalent:
(2) S is discriminated byS′ (i.e. for any finite set{s1, s2, . . . , sm} ⊆S there exist a homomorphism
φ:S →S′ withφ(si)≠ φ(sj) for alli̸=j).
Suppose c.s. semigroups S, S′ contain a subsemigroup T. We say that S is T-discriminated by S′ if for any finite set{s1, s2, . . . , sm} ⊆S there exist a homomorphismφ:S→S′ withφ(si)̸=φ(sj) for all
i̸=j and φ(t) =tfor each t∈T.
3. Coordinate Semigroups
Theorem 3.1. LetS′ = (G′,P′,Λ′, I′). A finitely generated semigroupS= (G,P,Λ, I)is the coordinate semigroup of an irreducible algebraic set over S′ iff there exists a subsemigroup T′ ⊆S′ isomorphic to
(G′,P,Λ, I), andG′ ⟨P⟩-discriminates G.
Proof. Let us prove the “only if” part of the theorem. Without loss of generality one can assume that the sandwich matrix P of S is (1,1)-normalized. Let E be the set of all idempotents of S (since S is finitely generated,E is finite), andG0 an arbitrary finite subset in G
P ={(1, p,1)|poccurs in the sandwich-matrix P},
Γ ={(1, g,1)|g∈G0}.
By Theorem 2.6, S is discriminated by S′, so there exists a homomorphism φ:S → S′ injective on
E ∪P ∪Γ. Let us denote the image of φ by ¯T. Following Remark 2.2, one can assume that ¯T is
isomorphic to the semigroup ( ¯G,P¯,Λ,¯ I), where¯ (1) the sandwich-matrix ¯Pis (1,1)-normalized;
(2) φ maps the first column ofS into the first column of ¯T and the first row of S is mapped into
the first row of ¯T.
By Lemma2.5, it holds (2.5) for some a∈G.¯
Sinceφis injective on the set of all idempotentsE ⊆S, we have ¯Λ = Λ, ¯I =I. One can renumber the
rows and columns of ¯T and the mappingsh, χ of the homomorphismφwill satisfyh(i) =i,χ(λ) =λ.
Clearly, the element a−1 defines the inner automorphism ga−1 = aga−1 of the group ¯G. Let ¯Ga denote the image of ¯Gvia this inner automorphism. Applying the inner automorphism to the elements
of sandwich-matrix ¯P, we obtain a new matrix ¯Pa.
Let Ta denote the c.s. semigroup ( ¯Ga,P¯a,Λ, I), and ψ: ¯T → Ta is an isomorphism between ¯T , Ta. Then for the homomorphism ϕ=ψ◦φthe equality (2.5) becomes
(3.1) ψ(ω(piλ)) =ph(i)χ(λ)=piλ.
Therefore, ¯Pa=P, and we have Ta= ( ¯Ga,P,Λ, I). Since we arbitrarily chose the set Γ, the group ¯Ga discriminates G and by (3.1) ϕdoes not change elements of P. Thus, the group ¯Ga ⟨P⟩-discriminates G. Since Ta ⊆T′= (G′,P,Λ, I)⊆S′ and ¯Ga is embedded into G′,G′ ⟨P⟩-discriminates G.
By condition, there exists a group⟨P⟩-homomorphismϕ:G→G′ withϕ(gi)̸=ϕ(gj) for anygi ̸=gj. Define a map ψ:S→T byψ((λ, g, k)) = (λ, ϕ(g), k).
The mapψ is a homomorphism, since
ψ((λ, g, k)(µ, h, l)) =ψ((λ, gpkµh, l)) = (λ, ϕ(gpkµh), l) = (λ, ϕ(g)ϕ(pkµ)ϕ(h), l) =
(λ, ϕ(g)pkµϕ(h), l),
and
ψ((λ, g, k))ψ((µ, h, l)) = (λ, ϕ(g), k)(µ, ϕ(h), l) = (λ, ϕ(g)pkµϕ(h), l).
For the elements ai, aj (i̸=j) we have
ψ(ai) = (λi, ϕ(gi), ki), ψ(ai) = (λj, ϕ(gj), kj).
If λi ̸= λj or ki ̸= kj it follows ψ(ai) ̸= ψi(aj). Otherwise (gi ̸= gj), by the choice of ϕ we obtain
ϕ(gi)̸=ϕ(gj), hence ψ(ai)̸=ψi(aj). □
Using Theorem 3.1, one can obtain the description of the coordinate semigroups of irreducible
alge-braic sets overFnab forn≥2.
Theorem 3.2. A finitely generated c.s. semigroup S is the coordinate semigroup of an irreducible algebraic set over Fnab (n≥2) iff it is isomorphic to a c.s. (G,P,Λ, I), where
(1)
Λ ={1,2, . . . , l}, I ={1,2, . . . , m}, l≤n, m≤n
(2)
(3.2) G= ⊕
i∈I,λ∈Λ
Ziλ⊕Zk
for some k≥0 (Ziλ is the isomorphic copy of the infinite cyclic group Z).
(3) P = (piλ) = (eiλ), where eiλ is the generating element of the maximal subgroup Ziλ ⊆ S. In
other words,G=⟨P⟩ ⊕Zk.
Proof. Suppose Fnab has a presentation (2.1), and S is the coordinate semigroup of an irreducible set over Fab
n . By Theorem 3.1, P is embedded into Pn, Λ ⊆ Λn, I ⊆ In and P = (piλ) = (eiλ) (i ∈ I, λ∈Λ).
Any group ⟨P⟩-discriminated byFN is of the form ⟨P⟩ ⊕Zk (see [8]), and we obtain thatS has the desired presentation (G,P,Λ, I).
Let us prove the “if” statement of the theorem. Let S = (G,P,Λ, I) with G,P,Λ, I defined above. Obviously,P is embedded intoPn. By Theorem 3.1, it is sufficient to prove Gis ⟨P⟩-discriminated by FN. By condition, G=⟨P⟩ ⊕Zk,FN =⟨P⟩ ⊕Zn. The group Zk is discriminated byZn (see [8]), and any homomorphism ψ ∈ Hom(Zk,Zn) can be extended to a homomorphism ψ′ ∈ Hom(G, FN) which
4. Algebraic Geometry over the Free c.s. Semigroups
In this section we show that all free semigroups{Fnab|n≥2}with abelian structural groups generate
the same algebraic geometry. Let us formalize this assertion as follows. Two c.s. semigroups S, S′
are geometrically equivalent if for any L-system S the coordinate semigroups ΓS(VS(S)), ΓS′(VS′(S)) are isomorphic. It follows that the geometrically equivalent semigroup S, S′ have the same set of the
coordinate algebras of algebraic sets.
We shall use the following test of geometrical equivalence (this result directly follows from Unifying
theorems of [4]).
Theorem 4.1. Finitely generated c.s. semigroupsS, S′ are geometrically equivalent iffS is separated by
S′ (i.e. for any distinct elementss1, s2 ∈S there exists a homomorphismφ:S →S′ withφ(s1)̸=φ(s2)) and S′ is separated by S.
Remark that the semigroupsFnabsatisfy the condition of Theorem4.1, since they are finitely generated.
Let us define a completely simple semigroupF0with a structural groupZ=⟨1⟩and a sandwich-matrix
P0 = (
0 0
0 1
)
Theorem 4.2. The semigroups Fnab,Fmab are geometrically equivalent for any m, n≥2.
Proof. Obviously, it is sufficient to prove that Fnab is geometrically equivalent to F0. According to Theorem4.1, we prove that Fnab,F0 separate each other.
Obviously F0 is separated by Fnab, since it is embedded into Fnab for each n≥ 2. Let us prove the
converse.
According to (2.2), the structural group and the sandwich-matrix ofFnab are
FN =⟨P⟩ ⊕Zn,
and Pn= (piλ) = (eiλ), where eiλ is the generator of the subgroupZiλ∼=Z. Obviously,
⟨Pn⟩=
⊕
1≤λ,i≤n Ziλ.
Let s1 = (λ, g, i), s2 = (µ, h, i) be two distinct elements of Fnab. Let us define a homomorphism ψ:Fnab→ F0 with ψ(s1)̸=ψ(s2).
Consider the next three cases.
(1) Let λ̸=µ and defineψ as follows
ψ((ν, f, k)) =
(1,0,1) ifν ̸=µ,
(2,0,1) ifν =µ
Roughly speaking,ψ maps all columns of the Rees matrix semigroupFnab into the first column
It is easy to check that ψ ∈Hom(Fnab,F0) and ψ(s1) ̸= ψ(s2), since s1, s2 are mapped into
the first and second rows respectively.
(2) The case i̸=j is similar to the previous one.
(3) Suppose λ=µ andi=j. The elementsg, hare uniquely represented as
g=p1+g′, h=p2+h′,
whereg′, h′ ∈Zn,p
1, p2∈ ⟨Pn⟩. We have exactly two cases:
(a) Suppose g′ ̸= h′. Since Zn is obviously separated by Z, there exists a homomorphism
φ∈Hom(Zn,Z) withφ(g′)≠ φ(h′). Letφ′ be the zero-extension ofφtoFN.
Thus,φ′(g)̸=φ′(h). Define a map ψ by
ψ((ν, f, k)) = (1, φ′(f),1).
It is directly checked thatψ∈Hom(Fnab,F0) and ψ(s1)̸=ψ(s2).
(b) Assume now that g′ =h′ and p1 ̸=p2. The elementsp1, p2 equal to the sums
pj =∑ i,λ
pjiλ, (1≤j≤2),
wherepjiλ∈Ziλ.
By the condition, there exists a pair of indexes i′ ∈ In, λ′ ∈ Λn such that p1i′λ′ ̸= p2i′λ′. The homomorphismφ:⟨P⟩ →Z by
φ(x) =
x, ifx∈Zi′λ′
0, otherwise
has the propertyφ(g)̸=φ(h).
Letφ′ be the zero-extension ofφtoFN. Define a map ψ:Fnab → F0 by
ψ((ν, f, k)) =
(2, φ′(f),2) ifν =λ′ and k=i′,
(2,0,1) ifk=i′,
(1,0,2) ifν=λ′,
(1,0,1) otherwise
Roughly speaking, ψ maps the i′-th column of Fnab to the second column of F0 and the
λ′-th row to the second row of F0. The another rows and columns ofFnab are mapped into
the first row and column ofF0.
It is directly checked thatψ∈Hom(Fnab,F0) and moreoverψ(s1)̸=ψ(s2).
Thus, the existence of a homomorphism ψ proves the separation ofFnab by F0. □
According to the main results of universal algebraic geometry, the variety var(A), the quasi-variety
of equations over A (see [4, 5] for more details). The following proposition contains the statements
about the universal classes generated by the c.s. semigroups Fnab.
Proposition 4.3. The universal classes of the semigroups Fmab,Fnab for n, m ≥ 2, n > m satisfy the inclusions
var(Fmab) =var(Fnab), qvar(Fmab) =qvar(Fnab), ucl(Fmab)⊂ucl(Fnab)
Proof. The first equality was proved in [10]. The second one follows from Theorem4.2and results of [5]. Since Fmab ⊆ Fnab, the results of model theory immediately gives ucl(Fmab)⊆ucl(Fnab) Let
φ: ∀x1∀x2· · · ∀xm2 m
2 ∧
i=1
x2i =xi →
∨
1≤i<j≤m2
xi =xj
be a universal formula of the languageLwhich states that a semigroup contains at mostm2idempotents. Clearly, φholds in Fmab not inFnab. Thus, the definition of the universal closure immediately gives the
strict inclusionucl(Fmab)⊂ucl(Fnab). □
Remark 4.4. It was proved in [7]that for free c.s. semigroups it holds var(Fm)⊂var(Fn) (n > m), and it directly implies the other strict inclusions: qvar(Fm)⊂qvar(Fn),ucl(Fm)⊂ucl(Fn).
5. Coordinate Semigroups. Equations with Constants
Let S′ be a c.s. semigroup and consider the extended language L(S′) = L ∪ {s′ | s′ ∈ S′}. The
new constant symbols correspond to all elements of S′. According to model theory, an L(S′)-term in
variables X={x1, x2, . . . , xn} is one of the following
(1) a variable xi;
(2) a constant symbols′ ∈S′;
(3) a product t(X)s(X) of twoL(S′)-terms t(X), s(X);
(4) (t(X))−1, wheret(X) is anL(S′)-term.
For example, the following expressionsx(s′y)−1, ((s′
1x)−1(ys′2)−1)−1 are L(S′)-terms. Obviously, the
class of L-terms is included into the class of all L(S′)-terms. The definitions of equations, algebraic
sets and coordinate semigroups in the language L(S′) are similar to the corresponding notions in the
languageL. Let us consider the differences between coordinate algebras in the languagesL andL(S′).
Recall that any constant s′ ∈ S′ is an L(S′)-term and s′1 ≁Y s′2 for any nonempty algebraic set Y
defined by L(S′)-equations overS′. Therefore, the constants form a subsemigroup isomorphic toS′ in
any coordinate semigroup ΓS′(Y). In other words, all coordinate semigroups of algebraic sets defined
by L(S′)-systems overS′ areS′-semigroups, i.e. they contain a fixed subsemigroup isomorphic to S′.
Proof. Let us prove the “only if” part of the statement. Since L(S′)-equations includes the class of
L-equations, the semigroupsS, S′ should satisfy the conditions of Theorem3.1. Therefore, there exists a subsemigroup T′ ⊆ S′, T′ = (G′,P,Λ, I) and S = (G,P,Λ, I). Since S is a S′-semigroup, Λ = Λ′, I =I′, and therefore P=P′.
The subgroupGS ={(1, g,1)|g∈G} ∼=G⊆S is mapped into a subgroup G′ ofS′ by any
homomor-phismψ∈HomS′(S, S′). Hence, the discrimination ofSimplies the discrimination of the groupGbyG′.
Since the subgroup G′S ={(1, g,1)|g ∈G′} ⊆GS ⊆S consists of constants, G′ should G′-discriminate
G.
The proof of the “if” part of the statement coincides with the corresponding statement of Theorem
3.1. □
Acknowledgments
The author was supported by the grant of Russian Science Foundation (project 17-11-01117).
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Artem N. Shevlyakov
Sobolev Institute of Mathematics Pevtsova st. 13 644099 Omsk, Russia, Omsk State Technical University, pr. Mira 11
644050 Omsk, Russia
